Ah, thank you SO MUCH! That article helped me learn a ton about Swing an AWT. I solved my problem by adding this:
public void initialize(){
JPopupMenu.setDefaultLightWeightPopupEnabled(false);
blah blah blah....
By adding that to the top of my code, the menu's always appear at the front. I've been stuck on this problem for a week, and I can't thank you enough.
I asked the same question at DreamInCode, but they banned me, blocked my IP, and filed an ISP complaint because my program is "against RuneScape's TOS". I mean, worry about yourself. They weren't very nice
Thanks again.